Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{q^2 - 25}{q + 5}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = q$ $ b = \sqrt{25} = 5$ So we can rewrite the expression as: $p = \dfrac{({q} + {5})({q} {-5})} {q + 5} $ We can divide the numerator and denominator by $(q + 5)$ on condition that $q \neq -5$ Therefore $p = q - 5; q \neq -5$